#8288355

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,15,20)(11,21,23,16)(12,18,13,17)(14,19,24,22)(26,28)(31,33)(32,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,19)(14,17)(15,23)(18,24)(20,21)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(2,5)(6,8)(9,12,15,13)(10,17,20,18)(11,22,23,19)(14,21,24,16)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,20,17)(11,22,23,19)(14,21,24,16)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,19)(14,17)(15,23)(18,24)(20,21);(1,2)(3,6)(4,5)(7,8)(9,14,15,24)(10,19,20,22)(11,17,23,18)(12,16,13,21)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8195551);